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4. Rational Exponents & Radical Functions
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Integrated Math 2

4.01 Simplifying algebraic radicals

Lesson
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Lesson

Concept summary

Radical expressions have many parts as shown in the following diagram:

Index

The number on a radical symbol that indicates which type of root it represents. For instance, the index on a cube root is 3. The index on a square root is usually not written, but would be 2

Radical

A mathematical expression that uses a root, such as a square root \sqrt{\quad}, or nth root \sqrt[n]{\quad}

Radicand

The value or expression inside the radical symbol

Perfect square

A number that is the result of multiplying two of the same integer

Perfect cube

A number that is the result of multiplying three of the same integer together

Radical expressions are written in simplified radical form if the radicand cannot be factored any further.

We can use the following facts to simplify rational expressions, for a, b \geq 0 and m,n positive integers,

\begin{aligned} \sqrt[n]{ab} &=\sqrt[n]{a} \sqrt[n]{b} \\ \sqrt[n]{a^n}&=a \\ a^{mn}&=\left(a^m\right)^n \end{aligned}

The same operations that can be applied when simplifying numerical radicals be applied when simplifying radicals with algebraic expressions.

Worked examples

Example 1

Simplify the following:\sqrt[3]{27x^3y^6}

Approach

To simplify a cube root we want to identify if the radicand has any factors that are perfect cubes. In this case, the numeric factor 27=3^3 and is a perfect cube. Looking at the algebraic factors we can see x^3 is obviously a perfect cube, and y^6=\left(y^2\right)^3 which is also a perfect cube.

Solution

Knowing this, we can simplify the original expression as follows:

\displaystyle \sqrt[3]{27x^3y^6}\displaystyle =\displaystyle \sqrt[3]{3^3} \sqrt[3]{x^3} \sqrt[3]{\left( y^2\right)^3}Product of powers
\displaystyle =\displaystyle 3xy^2Evaluating cube roots

This cannot be further simplified.

Example 2

Assuming that each variable represents a non-negative number, simplify the following:7\sqrt[4]{48b^5c^8}

Approach

To simplify a radical with index 4, we want to identify if the radicand has any factors of the form x^4. In this case, 48=3\left(16\right)=3\left(2^4\right) . Looking at the algebraic factors we can see that b^5=b^4b, so by splitting it into two factors, we will be able to take the root of b^4. Finally we have, \\c^8=\left(c^2\right)^4.

Solution

Knowing this, we can simplify the original expression as follows:

\displaystyle 7\sqrt[4]{48b^5c^8}\displaystyle =\displaystyle 7\sqrt[4]{3\left(2^4\right)b^4b\left(c^2\right)^4}Product of powers
\displaystyle =\displaystyle 7\sqrt[4]{2^4}\sqrt[4]{b^4}\sqrt[4]{b}\sqrt[4]{\left(c^2\right)^4}\sqrt[4]{3}Product of radicals
\displaystyle =\displaystyle 14bc^2 \sqrt[4]{3b}Evaluating fourth roots

Example 3

Simplify the following:\sqrt{\dfrac{4x^8}{25}}

Approach

To simplify the radical of two quotients we can rewrite it as the quotient of two radicals: \\\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}We can then express the radical in the numerator as the product of two radicals, one algebraic and one numeric.

Solution

\displaystyle \sqrt{\frac{4x^8}{25}}\displaystyle =\displaystyle \frac{\sqrt{4x^8}}{\sqrt{25}}Quotient of two radicals
\displaystyle =\displaystyle \dfrac{\sqrt{4}\sqrt{x^8}}{\sqrt{25}}Product of two radicals
\displaystyle =\displaystyle \frac{2x^4}{5}Evaluate square roots

Outcomes

M2.N.RN.A.1.C

Rewrite expressions involving radicals and rational exponents using the properties of exponents.

M2.MP1

Make sense of problems and persevere in solving them.

M2.MP6

Attend to precision.

M2.MP7

Look for and make use of structure.

M2.MP8

Look for and express regularity in repeated reasoning.

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